## Abstract

In this paper optimal upper bounds for the genus and the dimension of the graded components of the Hartshorne-Rao module of curves in projective n-space are established. This generalizes earlier work by Hartshorne [H] and Martin-Deschamps and Perrin [MDP]. Special emphasis is put on curves in P^{4}. The first main result is a so-called Restriction Theorem. It says that a non-degenerate curve of degree d ≥ 4 in P^{4} over a field of characteristic zero has a non-degenerate general hyperplane section if and only if it does not contain a planar curve of degreed - 1 (see Th. 1.3). Then, using methods of Brodmann and Nagel, bounds for the genus and Hartshorne-Rao module of curves in P^{n} with non-degenerate general hyperplane section are derived. It is shown that these bounds are best possible in a very strict sense. Coupling these bounds with the Restriction Theorem gives the second main result for curves in P^{4}. Then curves of maximal genus are investigated. The Betti numbers of their minimal free resolutions are computed and a description of all reduced curves of maximal genus in P^{n} of degree ≥ n + 2 is given. Finally, all pairs (d, g) of integers which really occur as the degree d and genus g of a non-degenerate curve in P^{4} are described.

Original language | English |
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Pages (from-to) | 695-724 |

Number of pages | 30 |

Journal | Mathematische Zeitschrift |

Volume | 229 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1998 |

## ASJC Scopus subject areas

- General Mathematics